2020 AMC 12B Problem 19

Question

Square

ABCD

in the coordinate plane has vertices at the points

A(1, 1),

B(-1, 1),

C(-1, -1),

and

D(1, -1).

Consider the following four transformations:

L,

a rotation of

90^\circ

counterclockwise around the origin;

R,

a rotation of

90^\circ

clockwise around the origin;

H,

a reflection across the

x

-axis; and

V,

a reflection across the

y

-axis.

Each of these transformations maps the square onto itself, but the positions of the labeled vertices will change. For example, applying

R

and then

V

would send the vertex

A

at

(1, 1)

to

(-1, -1)

and would send the vertex

B

at

(-1, 1)

to itself. How many sequences of

20

transformations chosen from

\{L, R, H, V\}

will send all of the labeled vertices back to their original positions? (For example,

R,

R,

V,

H

is one sequence of

4

transformations that will send the vertices back to their original positions.)

2^{37}

3 \cdot 2^{36}

2^{38}

3 \cdot 2^{37}

2^{39}

Answer: C

Concepts:transformation casework

Difficulty rating: 2000

Small hint:

The first

19

transformations can be anything; the last one is then forced to be the unique inverse

Big hint:

The forced last move must be one of the four allowed transformations; find the probability that it is

Solution:

Label the vertices

0,1,2,3

cyclically. Each allowed transformation has the form

j\mapsto \varepsilon j+\delta\pmod 4,

where

\varepsilon\in\{1,-1\}

and

\delta\in\{1,-1\}.

These four choices give exactly the two quarter-turns and the two stated reflections.

Under composition, the parity of

\delta

changes at every move. Thus a composition of

19

allowed transformations again has odd

\delta,

and so is one of the four allowed transformations. Its inverse is also allowed. Consequently every sequence of the first

19

moves has exactly one choice for the final move, giving

4^{19}=2^{38}

successful sequences.

Thus, the correct answer is C.